The (extended) coset leader and list weight enumerator

Transcription

1 The (extended) coset leader and list weight enumerator Relinde Jurrius (joint work with Ruud Pellikaan) Vrije Universiteit Brussel Fq11 July 24, 2013 Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

2 Extension codes Weight The number of nonzero coordinates of a vector. Linear [n, k] code Linear subspace C GF(q) n of dimension k. Elements are called (code)words, n is called the length. Generator matrix The rows of this k n matrix form a basis for C. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

3 Extension codes Weight The number of nonzero coordinates of a vector. Linear [n, k] code Linear subspace C GF(q) n of dimension k. Elements are called (code)words, n is called the length. Generator matrix The rows of this k n matrix form a basis for C. Extension code [n, k] code C GF(q m ) over some extension field GF(q m ) generated by the words of C. Generator matrix All the extension codes of C have the same generator matrix G. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

4 Coset leader weight enumerator Coset Translation of the code by a vector y GF(q) n. Weight The minimum weight of all vectors in the coset. Coset leader A vector of minimum weight in the coset. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

5 Coset leader weight enumerator Coset Translation of the code by a vector y GF(q) n. Weight The minimum weight of all vectors in the coset. Coset leader A vector of minimum weight in the coset. Extended coset leader weight enumerator The homogeneous polynomial counting the number of cosets of a given weight for all extension codes, notation: α C (X, Y, T ) = n α i (T )X n i Y i. i=0 Note that we have α C (X, Y, q m ) = α C GF(q m )(X, Y ). Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

6 List weight enumerator Extended list weight enumerator The homogeneous polynomial counting the number of coset leaders of a given weight for all extension codes, notation: λ C (X, Y, T ) = n λ i (T )X n i Y i. i=0 Note that we have λ C (X, Y, q m ) = λ C GF(q m )(X, Y ). Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

7 List weight enumerator Extended list weight enumerator The homogeneous polynomial counting the number of coset leaders of a given weight for all extension codes, notation: λ C (X, Y, T ) = n λ i (T )X n i Y i. i=0 Note that we have λ C (X, Y, q m ) = λ C GF(q m )(X, Y ). In general, we have α i (T ) λ i (T ), with equality iff all cosets of weight i have a unique coset leader. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

8 Why do we study this? The extended coset leader weight enumerator is interesting because: Determines the probability of correct decoding in coset leader decoding. Determines the average of changed symbols in steganography (information hiding). Not determined by the extended weight enumerator. The extended list weight enumerator is interesting because: Determines the size of lists in list decoding. Determines the probability of correct decoding in list decoding.... and of course because they are invariants of linear codes. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

9 Hyperplane arrangements and projective systems Arrangement of hyperplanes n-tuple of hyperplanes in GF(q) k. Essential arrangement Intersection of all hyperplanes is {0}, hyperplanes are in PG(k 1, q). Projective system n-tuple of points in PG(k 1, q). Projective systems are the geometric duals of hyperplane arrangements. Both induce the same geometric lattice. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

10 Hyperplane arrangements and projective systems Arrangement of hyperplanes n-tuple of hyperplanes in GF(q) k. Essential arrangement Intersection of all hyperplanes is {0}, hyperplanes are in PG(k 1, q). Projective system n-tuple of points in PG(k 1, q). Projective systems are the geometric duals of hyperplane arrangements. Both induce the same geometric lattice. Columns of a generator matrix G of a linear [n, k] code form a hyperplane arrangement / projective system. One-to-one correspondence between equivalence classes. Independent of choice of G, so notation: A C or P C. Also valid over an extension field GF(q m ). Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

11 Determination of coset weights Parity check matrix (n k) n matrix H such that GH T = 0. Syndrome of y The vector s = Hy T, zero for codewords. Syndrome weight Minimal number of columns whose span contains s. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

12 Determination of coset weights Parity check matrix (n k) n matrix H such that GH T = 0. Syndrome of y The vector s = Hy T, zero for codewords. Syndrome weight Minimal number of columns whose span contains s. Isomorphism between cosets and syndromes, because H(y + c) T = Hy T + Hc T = Hy T. Syndrome weight is equal to corresponding coset weight (weight of coset leader). α i is the number of vectors that are in the span of i columns of H but not in the span of i 1 columns of H. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

13 Determination of coset weights Example The extended coset leader weights are given by The code itself. α 0 (T ) = 1 The [7, 4] binary Hamming code has parity check matrix H = This is a projective system of seven points in the projective plane. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

14 Determination of coset weights Example The extended coset leader weights are given by Seven projective points. The [7, 4] binary Hamming code has parity check matrix H = This is a projective system of seven points in the projective plane. α 1 (T ) = 7(T 1) Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

15 Determination of coset weights Example The extended coset leader weights are given by The [7, 4] binary Hamming code has parity check matrix H = This is a projective system of seven points in the projective plane. α 2 (T ) = 7(T 1)(T 2) (T + 1) 3 extra points on 7 projective lines. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

16 Determination of coset weights Example The extended coset leader weights are given by The [7, 4] binary Hamming code has parity check matrix H = This is a projective system of seven points in the projective plane. α 3 (T ) = (T 1)(T 2)(T 4) α 0 (T ) + α 1 (T ) + α 2 (T ) + α 3 (T ) = T 3 total number of cosets. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

17 Determination of coset weights This kind of inclusion/exclusion counting is formalized by the geometric lattice associated to a projective system and its Möbius function. Example (continued) The geometric lattice associated to the [7, 4] binary Hamming code is visualized by abcdefg abc adg aef beg bdf cde cfg a b c d e f g Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

18 Determination of coset weights Example 1 coset with 3 cosets with 3 coset leaders 2 leaders each Projective systems with equal geometric lattices may have different coset leader weight enumerators! Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

19 Derived code Start with [n, k] code. Consider the projective system P C. Look at all hyperplanes spanned by k 1 points of P C. (Ignore k 1 points that span spaces of lower dimension.) Remove (multiple) copies of hyperplanes. These hyperplanes form an arrangement A. The derived code D(C) is the code such that A = A D(C). Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

20 Derived code Example d a f 2 c g b e abcd ab ac ad bc bd cd a b c d Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

21 Derived code Example d a f 2 c g b e V Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

22 Extended coset leader weight enumerator The lattice of P C, upside-down, is contained in the lattice of A D(C). This gives an injection ψ : L(P C ) L(A D(C) ). All elements that are not in the image ψ(l(p C )) should be counted similar to the largest element below it that is in ψ(l(p C )). Therefore, define r (x) = max{r(y) : y ψ(l(p C )), y x}. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

23 Extended coset leader weight enumerator The lattice of P C, upside-down, is contained in the lattice of A D(C). This gives an injection ψ : L(P C ) L(A D(C) ). All elements that are not in the image ψ(l(p C )) should be counted similar to the largest element below it that is in ψ(l(p C )). Therefore, define r (x) = max{r(y) : y ψ(l(p C )), y x}. Theorem The extended coset leader weight enumerator is equal to α C (X, Y, T ) = µ(x, y)t n k r(y) X k+r (x) Y n k r (x). x,y L(A D(C) ) Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

24 Summary The extended coset leader weight enumerator is an important invariant of linear codes. Determining coset weights is equivalent to counting points in spans of points. Counting points can be formalized by using the geometric lattice of the derived code. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

25 Further questions Does the extended coset leader weight enumerator determine the extended weight enumerator? Can we define a derived lattice? Taking D(D(D( (C) ))) eventually gives all hyperplanes in PG(k 1, q). How fast? Dependencies between dependencies are known as second order syzygies in computational geometry. Can this interpretation help? Can we determine α C (X, Y, T ) for concrete classes of codes? (For example: generalized Reed-Solomon codes) Can we classify codes using their coset leader weight enumerator?... Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16

26 Thank you for your attention. Relinde Jurrius (VUB) Coset leader weight enumerator Fq11 July 24, / 16